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A002117 Decimal expansion of zeta(3) = sum(m>=1, 1/m^3 ).
(Formerly M0020)
59
1, 2, 0, 2, 0, 5, 6, 9, 0, 3, 1, 5, 9, 5, 9, 4, 2, 8, 5, 3, 9, 9, 7, 3, 8, 1, 6, 1, 5, 1, 1, 4, 4, 9, 9, 9, 0, 7, 6, 4, 9, 8, 6, 2, 9, 2, 3, 4, 0, 4, 9, 8, 8, 8, 1, 7, 9, 2, 2, 7, 1, 5, 5, 5, 3, 4, 1, 8, 3, 8, 2, 0, 5, 7, 8, 6, 3, 1, 3, 0, 9, 0, 1, 8, 6, 4, 5, 5, 8, 7, 3, 6, 0, 9, 3, 3, 5, 2, 5, 8, 1, 4, 6, 1, 9, 9, 1, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sometimes called Apery's constant.

"A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apery succeeded in proving that Zeta(3) is irrational. In Chapter 8 we pointed out that the probability that two random integers are relatively prime is 6/Pi^2, which is 1/Zeta(2). This generalizes to: The probability that k random integers are relatively prime is 1/Zeta(k) ... ." [Stan Wagon]

In 2001 Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See the Rivoal link for further information and references.

The reciprocal of this constant is the probability that three integers chosen randomly using uniform distribution are relatively prime. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2005

Also the value of zeta(1,2), the double zeta-function of arguments 1 and 2. - R. J. Mathar, Oct 10 2011

REFERENCES

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 40-53

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.

Hardy and Wright, 'An Introduction to the Theory of Numbers' pp. 47,268-269.

Yasuyuki Kachi and Pavlos Tzermias, Infinite products involving zeta(3) and Catalan's constant, Journal of Integer Sequences, Vol. 15 (2012), #12.9.4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stan Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 354.

Yaglom and Yaglom, 'Challenging Mathematical Problems with Elementary Solutions' ex. 92-93

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20002

T. Amdeberhan, Faster and Faster convergent series for zeta(3), arXiv:math.CO/9084126

Dr. Math, Probability of Random Numbers Being Coprime

P. Bala, New series for old functions

J. Borwein and D. Bradley, Empirically determined Ap'ery-like formulae for zeta(4n+3)

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

X. Gourdon and P. Sebah, The Apery's constant:zeta(3)

W. Janous, Around Apery's constant, J. Inequ. Pure Appl. Math. 7 (2006) vol. 1, #35

M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method, arXiv:math.CA/0405592

F. M. S. Lima, A simple approximate expression for the Ape'ry's constant accurate to 21 digits, Oct 14, 2009 [From Jonathan Vos Post, Oct 14 2009]

R. J. Mathar, Yet another table of integrals, arXiv:1207.5845 [math.CA]

S. D. Miller, An Easier Way to Show zeta(3) is Irrational

Simon Plouffe, Zeta(3) or Apery's constant to 2000 places

A. van der Poorten, A Proof that Euler Missed

Tanguy Rivoal, Irrationality of the zeta Function on Odd Integers

G. Villemin's Almanach of Numbers, Apery's Constant(Text in French)

S. Wedeniwski, The value of zeta(3) to 1000000 places [Gutenberg Project Etext]

S. Wedeniwski, Plouffe's Inverter, Apery's constant to 128000026 decimal digits

S. Wedeniwski, The value of zeta(3) to 1000000 decimal digits

Eric Weisstein's World of Mathematics, Apery's Constant

Eric Weisstein's World of Mathematics, Relatively Prime

Wikipedia, Riemann zeta function

H. Wilf, Accelerated series for universal constants, by the WZ method

Wadim Zudilin, An elementary proof of Apery's theorem

Index entries for zeta function.

FORMULA

Lima gives an approximation to zeta(3) as (236*ln(2)^3)/197 - 283/394*pi*ln(2)^2 + 11/394*pi^2*ln(2) + 209/394*ln(sqrt(2) + 1)^3 - 5/197 + (93*Catalan*pi)/197. [From Jonathan Vos Post, Oct 14 2009] [Corrected by Wouter Meeussen, Apr 04 2010]

Zeta(3) = 5/2*integral(x=0..2*log((1+sqrt(5))/2), x^2/(exp(x)-1) ) + 10/3*(log((1+sqrt(5))/2))^3. [from Seiichi Kirikami, Fri Aug 12 2011]

Zeta(3) = -4/3*integral(x=0..1) log(x)/x*log(1+x) = integral(x=0..1) log(x)/x*log(1-x) = -4/7*integral(x=0..1) log(x)/x*log((1+x)/(1-x)) = 4*integral(x=0..1) 1/x*log(1+x)^2 = 1/2*integral(x=0..1) 1/x*log(1-x)^2 = -16/7*integral(x=0..Pi/2) x*log(2*cos(x)) = -4/Pi*integral(x=0..Pi/2) x^2*log(2*cos(x)). [Jean-Fran├žois Alcover, Apr 02 2013, after R. J. Mathar]

From Peter Bala, Dec 04 2013: (Start)

zeta(3) = 16/7*sum {k even} (k^3 + k^5)/(k^2 - 1)^4.

zeta(3) - 1 = sum {k >= 1} 1/(k^3 + 4*k^7) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 -...- (n - 1)^6/((2*n - 1)*(n^2 - n + 5) - ...))))) (continued fraction).

More generally, there is a sequence of polynomials P(n,x) (of degree 2*n) such that

zeta(3) - sum {k = 1..n} 1/k^3 = sum {k >= 1} 1/( k^3*P(n,k-1)*P(n,k) ) = 1/((2*n^2 + 2*n + 1) - 1^6/(3*(2*n^2 + 2*n + 3) - 2^6/(5*(2*n^2 + 2*n + 7) - 3^6/(7*(2*n^2 + 2*n + 13) - ...)))) (continued fraction). See A143003 and A143007 for details.

Series acceleration formulas:

zeta(3) = 5/2*sum {n >= 1} (-1)^(n+1)/( n^3*binomial(2*n,n) )

= 5/2*sum {n >= 1} P(n)/( (2*n(2*n - 1))^3*binomial(4*n,2*n) )

= 5/2*sum {n >= 1} (-1)^(n+1)*Q(n)/( (3*n(3*n - 1)*(3*n - 2))^3*binomial(6*n,3*n) ), where P(n) = 24*n^3 + 4*n^2 - 6*n + 1 and Q(n) = 9477*n^6 - 11421*n^5 + 5265*n^4 - 1701*n^3 + 558*n^2 - 108*n + 8 (Bala, section 7). (End)

EXAMPLE

1.2020569031595942853997...

MATHEMATICA

RealDigits[ N[ Zeta[3], 100] ] [ [1] ]

PROG

(PARI) { default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002117.txt", n, " ", d)); } \\ Harry J. Smith, Apr 19 2009

(Maxima) fpprec : 100$ ev(bfloat(zeta(3)))$ bfloat(%); // Martin Ettl, Oct 21 2012

CROSSREFS

Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, A013677, A059956 (6/Pi^2), A084225; A084226.

Cf. A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8; A233091: 7*zeta(3)/8.

Cf. A143003, A143007.

Sequence in context: A011420 A035686 A037228 * A042970 A158327 A136581

Adjacent sequences:  A002114 A002115 A002116 * A002118 A002119 A002120

KEYWORD

cons,nonn,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from David W. Wilson. Additional comments from Robert G. Wilson v, Dec 08 2000

Quotation from Stan Wagon corrected by N. J. A. Sloane on Dec 24 2005. Thanks to Jose Brox for noticing this error.

STATUS

approved

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Last modified April 20 14:09 EDT 2014. Contains 240806 sequences.